Ill] IN VARIOUS ORGANISMS 159 



the same. Growth keeps caUing our attention to its own com- 

 plexity. We see it in the rates of growth which change with age 

 or season, which vary from one hmb to another,; in the influence 

 of peace and plenty, of war and famine ; not least in those composite 

 populations whose own parts aid or hamper one another, in any 

 form or aspect of the struggle for existence. So we come to the 

 differential equations, easy to frame, more difficult to solve, easy in 

 their first steps, hard and very powerful later on, by which Lotka 

 and Volterra have shewn how to apply mathematics to evolutionary 

 biology, but which he just outside the scope of this book*. 



An important element in a population, and one seldom easy to 

 define, is its age-composition. It may vary one way or the other; 

 for the diminution of a population may be due to a decrease in the 

 birth-rate, or to an increasing mortality among the old. A remark- 

 able instance is that of the food-fishes of the North Sea. Their 

 birth-rate is so high that the very young fishes remain, to all 

 appearance, as numerous as ever; those somewhat older are fewer 

 than before, and the old dwindle to a fraction of what they were 

 wont to be. 



The rate of growth in other organisms 



The rise and fall of growth-rate, the acceleration followed by 

 retardation which finds expression in the S^shaped curve, are seen 

 alike in the growth of a population and of an individual, and in 

 most things which have a beginning and an end. But the law of 

 large numbers smooths the population-curve; the individual Hfe 

 draws attention to its own ups and downs; and the characteristic 

 sigmoid curve is only seen in the simpler organisms, or in parts or 

 "phases" of the more complex lives. We see it at its simplest in 

 the simple growth-cycle, or single season, of an annual plant, which 

 cycle draws to its end at flowering; and here not only is the curve 

 simple, but its amplitude may sometimes be very large. The giant 

 Heracleum and certain tall varieties of Indian corn grow to twelve feet 



* See (int. al.) A. J. Lotka, Elements of Physical Biology, Baltimore, 1925; 

 Theorie analytique des associations biologiques, Paris, 1934; Vito Volterra, Lemons 

 sur la theorie mathematique de la lutte pour la vie, 1931; Volterra et U. d'Ancona, 

 Les associations biologiques au point de vue mathematique, 1935; V. A. Kostitzin, 

 op. ci7.;\ etc 



